S. P. Goldman

Feasibility of a fast inverse dose optimization algorithm for IMRT via matrix inversion without negative beamlet intensities

A fast optimization algorithm is very important for inverse planning of intensity modulated radiation therapy (IMRT), and for adaptive radiotherapy of the future. Conventional numerical search algorithms such as the conjugate gradient search, with positive beam weight constraints, generally require numerous iterations and may produce suboptimal dose results due to trapping in local minima. A direct solution of the inverse problem using conventional quadratic objective functions without positive beam constraints is more efficient but will result in unrealistic negative beam weights. We present here a direct solution of the inverse problem that does not yield unphysical negative beam weights. The objective function for the optimization of a large number of beamlets is reformulated such that the optimization problem is reduced to a linear set of equations. The optimal set of intensities is found through a matrix inversion, and negative beamlet intensities are avoided without the need for externally imposed ad-hoc constraints. The method has been demonstrated with a test phantom and a few clinical radiotherapy cases, using primary dose calculations. We achieve highly conformal primary dose distributions with very rapid optimization times. Typical optimization times for a single anatomical slice (two dimensional) (head and neck) using a LAPACK matrix inversion routine in a single processor desktop computer, are: 0.03 s for 500 beamlets; 0.28 s for 1000 beamlets; 3.1 s for 2000 beamlets; and 12 s for 3000 beamlets. Clinical implementation will require the additional time of a one-time precomputation of scattered radiation for all beamlets, but will not impact the optimization speed. In conclusion, the new method provides a fast and robust technique to find a global minimum that yields excellent results for the inverse planning of IMRT.

1/Z expansion calculation of the Bethe logarithm for the ground state Lamb shift of two-electron ions

The leading two terms in the 1/Z expansion of the two-electron Bethe logarithm are calculated by the application of a new finite basis set method. The results can be expressed in the form ln epsilon (1s2 1S)=ln(19.77(Z-0.0063)2). The high-Z behaviour appears to differ from that of a previous variational calculation by Aashamar and Austvik (1976).

Extended relativistic representation of spin-1/2 particles

The variational approach to the Dirac-Coulomb Hamiltonian is extended to include simultaneously real electron and positron states. The spurious root previously obtained using finite-basis sets for states with kappa >0 is now clearly identified as the variational ground state of a negative-energy positron, resulting in a variational spectrum that is symmetric around E=0 and free of spurious roots. Moreover, a criterion for the accuracy of the variational results is obtained in terms of the (unphysical) positron component of the electron wavefunction. This approach is then further extended non-trivially to mix the non-optimized variational electron and positron states in the presence of a Coulomb potential. Exact solutions are presented. It is found that shifts in the non-optimized variational Dirac energy levels are obtained, expressed in terms of an arbitrary angle. This property is used to avoid variational collapse with any basis set and, as a consequence, the projector over positive-energy states is derived in a simple analytic form.